Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model
A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remain...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2022 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2022
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211540 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model. Ian Marquette and Christiane Quesne. SIGMA 18 (2022), 005, 24 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862684274792071168 |
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| author | Marquette, Ian Quesne, Christiane |
| author_facet | Marquette, Ian Quesne, Christiane |
| citation_txt | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model. Ian Marquette and Christiane Quesne. SIGMA 18 (2022), 005, 24 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remain to be studied. We present a set of six operators {⁺⁻, ⁺⁻, ⁺⁻} that can be combined to build a (3) hidden algebra. The latter can be embedded in an (6) algebra, as well as in an (1/6) superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences.
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| first_indexed | 2026-03-17T09:27:48Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211540 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T09:27:48Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Marquette, Ian Quesne, Christiane 2026-01-05T12:30:23Z 2022 Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model. Ian Marquette and Christiane Quesne. SIGMA 18 (2022), 005, 24 pages 1815-0659 2020 Mathematics Subject Classification: 81Q05; 81Q60; 81R12; 81R15 arXiv:2010.15276 https://nasplib.isofts.kiev.ua/handle/123456789/211540 https://doi.org/10.3842/SIGMA.2022.005 A shape invariant nonseparable and nondiagonalizable three-dimensional model with quadratic complex interaction was introduced by Bardavelidze, Cannata, Ioffe, and Nishnianidze. However, the complete hidden symmetry algebra and the description of the associated states that form Jordan blocks remain to be studied. We present a set of six operators {⁺⁻, ⁺⁻, ⁺⁻} that can be combined to build a (3) hidden algebra. The latter can be embedded in an (6) algebra, as well as in an (1/6) superalgebra. The states associated with the eigenstates and making Jordan blocks are induced in different ways by combinations of operators acting on the ground state. We present the action of these operators and study the construction of an extended biorthogonal basis. These rely on establishing various nontrivial polynomial and commutator identities. We also make a connection between the hidden symmetry and the underlying superintegrability property of the model. Interestingly, the integrals generate a cubic algebra. This work demonstrates how various concepts that have been applied widely to Hermitian Hamiltonians, such as hidden symmetries, superintegrability, and ladder operators, extend to the pseudo-Hermitian case with many differences. I. Marquette was supported by the Australian Research Council Future Fellowship FT180100099. C. Quesne was supported by the Fonds de la Recherche Scientifique- FNRS under Grant Number 4.45.10.08. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model Article published earlier |
| spellingShingle | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model Marquette, Ian Quesne, Christiane |
| title | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model |
| title_full | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model |
| title_fullStr | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model |
| title_full_unstemmed | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model |
| title_short | Ladder Operators and Hidden Algebras for Shape Invariant Nonseparable and Nondiagonalizable Modelswith Quadratic Complex Interaction. II. Three-Dimensional Model |
| title_sort | ladder operators and hidden algebras for shape invariant nonseparable and nondiagonalizable modelswith quadratic complex interaction. ii. three-dimensional model |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211540 |
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